Exponential Decay Quick Check

Question

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Question
A local bakery opened with new equipment worth $80,000. Ten years later, the owners’ accountants told them that their equipment had depreciated at a rate of 6 percent per year. Which equation would you use to create a sequence modeling this situation?(1 point)
Responses

an=80,000(0.06)n−1
a subscript n baseline equals 80,000 left parenthesis 0.06 right parenthesis superscript n minus 1 baseline

an=80,000(0.06)n
a subscript n baseline equals 80,000 left parenthesis 0.06 right parenthesis superscript n baseline

an=80,000(0.94)n
a subscript n baseline equals 80,000 left parenthesis 0.94 right parenthesis superscript n baseline

an=80,000(0.94)n−1
a subscript n baseline equals 80,000 left parenthesis 0.94 right parenthesis superscript n minus 1 baseline
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Answer 1

To model the exponential decay of the bakery's equipment, we need to consider that the equipment depreciates at a rate of 6% per year. This means that each year, the value of the equipment retains 94% of its value from the previous year (since 100% - 6% = 94%).

The general formula for exponential decay is given by:

\[ a_n = a_0(1 - r)^n \]

Where:

• \( a_n \) is the value after \( n \) years,
• \( a_0 \) is the initial value (in this case, $80,000),
• \( r \) is the decay rate (0.06 for 6%),
• \( n \) is the number of years.

Substituting into the formula gives us:

\[ a_n = 80,000(0.94)^n \]

Thus, the correct equation to model the depreciation of the bakery's equipment is:

\( a_n = 80,000(0.94)^n \)

Therefore, the correct response from the given options is:

\( a_n = 80,000(0.94)^n \).